On a February afternoon in 1866, a 44-year-old New Hampshire-born invalid named Mary Baker Patterson — later Eddy — slipped on a patch of ice in Lynn, Massachusetts, and was carried home with what her doctors said were probably terminal injuries. Three days later, she opened a Bible to a passage about Jesus healing a paralytic, by her own account closed it, and walked. The experience set her on a thirty-year project of trying to write down what had happened. The result, published in 1875, was Science and Health with Key to the Scriptures. The text was a dense, idiosyncratic theological argument that the material world was an illusion, that physical disease was an error of belief, and that prayer alone — properly applied — could correct it.

Eddy chartered the Church of Christ (Scientist) with twenty-six followers in 1879, reorganized it as the Church of Christ, Scientist in 1892, and oversaw the building of the Mother Church on the Back Bay in Boston, completed in 1894. The denomination forbade most medical treatment. It still does, with carefully circumscribed exceptions for childbirth, dentistry, and broken bones. Several decades of legal cases involving Christian Science children dying of treatable conditions have gradually narrowed the legal exemptions in U.S. statute.

In 1908, at age 87, Eddy founded The Christian Science Monitor as a non-proselytizing daily newspaper to counter what she saw as the sensationalism of the Hearst press. The paper has won seven Pulitzer Prizes between 1950 and 2002, gone through several near-extinction events, and now publishes mostly online. It has consistently been more respected than the church it funds. Eddy died in 1910, leaving an estate valued at $1.5 million. Membership in the church peaked at roughly 270,000 in 1936 and has been falling for nearly a century.

On June 7, 1742, the Prussian-born mathematician Christian Goldbach wrote a letter to Leonhard Euler with a casual conjecture: every integer greater than 2 could be written as the sum of three primes. Euler, in his reply, restated it: every even integer greater than 2 is the sum of two primes. (4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 100 = 47 + 53; and so on for every even number you've ever cared about.) Euler called it "a completely certain theorem, although I cannot prove it." Two hundred and eighty-three years later, neither can anyone else.

The conjecture is one of the oldest unsolved problems in number theory and the simplest to state. The shortest research papers responding to it run to 200 pages. Computational verification has been thorough: the conjecture has been confirmed for every even integer up to 4 × 10¹⁸, with no counterexamples and no near-misses. Most large even numbers can be written as a sum of two primes in many different ways, and the count goes up roughly as expected from heuristic prime-density arguments. The proof remains out of reach.

A softer version is partially solved. The weak Goldbach conjecture asserts that every odd integer greater than 5 is the sum of three primes. The Peruvian mathematician Harald Helfgott proved the weak conjecture in 2013 — a 130-page paper, cleaning up an approach that had been waiting since the 1930s for enough computer time to fill in the inevitable last finite gap. The strong conjecture, the original Goldbach, would imply the weak one, but not vice versa. The closest known result on the strong conjecture is from the Chinese mathematician Chen Jingrun in 1973: every sufficiently large even number is the sum of either two primes or one prime and a product of two primes. "Sufficiently large" still includes infinitely many cases.

The Cathars were a dualist Christian movement that spread through southern France, northern Italy, and parts of the Rhineland between roughly 1100 and 1300. Their theology held that the visible material world had been created not by the good God of the Gospels but by a malign creator, Rex Mundi, the King of the World, and that salvation lay in renouncing material existence. They rejected the Catholic clerical hierarchy, the sacraments, and most of the institutional church, and conducted their own initiation, the consolamentum, by laying-on of hands rather than water baptism. The Languedoc region of southern France was substantially tolerant of them through the 12th century — Occitan-speaking, urbanized, governed by counts who didn't always answer to Paris.

In January 1208 a Cistercian papal legate, Pierre de Castelnau, was murdered in the Rhône Valley in circumstances that pointed to Count Raymond VI of Toulouse, a Cathar sympathizer. Pope Innocent III responded by calling a crusade. Twenty thousand or so Catholic knights and footmen marched south from Lyon under the command of the abbot Arnaud Amalric. The first major target was Béziers, a Languedoc city of mixed Catholic and Cathar population, in July 1209. The chronicle attributed to Caesarius of Heisterbach reports that when soldiers asked the legate how to distinguish the faithful from the heretics, Arnaud Amalric replied, "Caedite eos. Novit enim Dominus qui sunt eius" — "Kill them all. The Lord will know his own." Whether the words were actually said is debated; the conduct was not. Béziers was burned and its inhabitants slaughtered.

The crusade ran for twenty years, ending with the Treaty of Paris on April 12, 1229. Modern estimates put the total Cathar dead across the period at roughly 200,000. Raphael Lemkin, who in 1944 coined the word genocide, cited the Albigensian Crusade as one of his foundational examples.

An aperiodic tiling is a way of covering the plane with copies of a small set of tiles such that the resulting pattern never repeats — slide it, and the alignment fails. The 1960s mathematical question was whether such tilings existed at all. Robert Berger answered yes in 1966, with a set of about 20,000 tiles. Through the late 1960s and early 1970s, the count was reduced. In 1974 the Oxford mathematical physicist Roger Penrose published a set of just two tiles — a "kite" and a "dart," or alternatively a fat and thin rhombus — that could tile the plane in an infinity of ways, but never periodically. The arrangement showed an apparent five-fold rotational symmetry that classical crystallography had said was impossible.

The pattern's afterlife crossed disciplines. In 1982, the materials scientist Dan Shechtman observed an aluminum-manganese alloy diffracting electrons in a sharp ten-fold pattern that looked exactly like what Penrose tilings predicted. Crystallographers initially refused to publish his result. Linus Pauling famously called him "a quasi-scientist." Shechtman won the 2011 Nobel Prize in Chemistry for the discovery, by which point quasicrystals had been found in nature in a Russian-meteorite Khatyrkite sample.

The oddest historical wrinkle came in 2007. The Harvard physicist Peter Lu and the Princeton physicist Paul Steinhardt analyzed Iranian girih tile-work — the geometric pattern tradition decorating Islamic architecture — and showed that the spandrels of the Darb-e Imam shrine in Isfahan, built in 1453, had been laid out as an effectively perfect Penrose tiling. The fifteenth-century Persian master tilers had cracked the math more than five hundred years before Penrose. They left no recorded explanation of how.

Patriarch Nikon of Moscow took office in 1652 and immediately began a project of aligning Russian Orthodox practice with the Greek liturgical norms used at Constantinople. The changes were small in print and enormous in implication for the people whose lives they governed. The sign of the cross, Russians had been making with two fingers — index and middle, with the thumb folded across the ring and little finger — to symbolize the two natures of Christ. Nikon prescribed three fingers, in the Greek style, to represent the Trinity. Liturgical processions were redirected. The spelling of Iisus was added to the older Isus. The number of prosphora (offering loaves) used during the Eucharist was changed.

The response was a popular religious crisis. Many Russian peasants and clergy refused. The Great Moscow Synod of 1666–67 formally anathematized the dissenters, declaring them schismatics — raskolniki. Imperial enforcement followed. The Solovetsky Monastery, on an island in the White Sea, expelled its reformist abbot and held out under armed siege from Tsar Alexei's troops between 1668 and January 1676; when the walls were breached, almost all of its remaining several hundred monks were executed.

The most famous Old Believer leader, the archpriest Avvakum, refused to recant under multiple imprisonments and exiles. He was kept for fifteen years in a frozen dugout at Pustozersk on the Arctic coast, where he wrote a startling first-person spiritual autobiography that survives. On April 14, 1682, Avvakum and three companions were burned at the stake. The Old Believer communities, splintered into a dozen subgroups, persisted underground. Demographers estimated 10 to 20 million Old Believers in the early twentieth century. The Russian Orthodox Church formally lifted the anathema only in 1971.

Heaven's Gate started in 1974 as a partnership between Marshall Herff Applewhite, a 43-year-old former music professor in Texas, and Bonnie Lu Nettles, a 47-year-old Houston nurse with a background in theosophy. The two began traveling around the United States preaching what they described as a UFO-based gospel: human bodies were vehicles, the soul could be lifted to a higher "Next Level," and a flying saucer would arrive on cue to pick up the prepared. They called themselves "Bo and Peep," then "The Two," and finally "Do and Ti." Through the late 1970s and 1980s, they traveled with small groups of followers in itinerant communal living arrangements, requiring chastity, communal property, and gradual departure from family and prior identity.

Nettles died of cancer in 1985, which forced a theological revision: if the saucer hadn't come to take her body, the body must not be the relevant vehicle. Applewhite reframed the doctrine to allow for soul-only ascension. The group settled in Rancho Santa Fe, California, in 1996, supported themselves running a small website-development business under the name Higher Source, and spent the autumn watching the approach of the comet Hale-Bopp.

Applewhite and 38 of his followers killed themselves in coordinated waves across March 22 to 24, 1997. Sheriff's deputies found the bodies on March 26. Each of the 39 was wearing identical black trousers, a black long-sleeved shirt, brand-new black-and-white Nike Decades sneakers, and a square purple cloth folded over their head. Each had $5.75 in cash and a roll of quarters in a pocket. Each had drunk applesauce mixed with phenobarbital and vodka, then tied a plastic bag over their head. Their farewell video, posted to the Heaven's Gate website, said they were "shedding" their containers to board a vehicle traveling in Hale-Bopp's tail. The website is, eccentrically, still online.

In the spring of 1981, the Raiders of the Lost Ark shoot was three months in and based out of Kairouan, Tunisia. The scene to be filmed that morning had three pages of dialogue and stage direction: a Cairo street, a turbaned swordsman appears, threatens Marion, and engages Indiana Jones in an elaborate whip-versus-scimitar duel that the script blocked out for roughly three days of shooting. The swordsman, played by British stuntman Terry Richards, had spent months training for it.

Harrison Ford was sick. Three months of dust and Tunisian heat had left him with dysentery — unable, in his own description, to stay away from his trailer for longer than one magazine of film. Ten minutes was about all he had.

He walked up to Steven Spielberg that morning and said, in the version both have repeated since, "Steven, why don't we just shoot this sumb***h?" Spielberg said he had been thinking the same thing.

In one take, the camera holds on Richards doing his sword work. Indy looks at the blade, looks at the camera with mild exhaustion, draws his revolver, and shoots him. Cut. At the first preview screening, audiences laughed for about ten seconds straight. The scene is one of the most quoted moments in the film and probably its most beloved improvisation.

Both Ford and Spielberg have confirmed the dysentery story on the record. Richards, the swordsman, was reportedly disappointed but professional about it. The film opened on June 12, 1981.

Sofia Vasilyevna Kovalevskaya was a Russian mathematician born in 1850 who could not, under Russian law of the 1860s, attend university. The available workaround was a fictitious marriage to a man willing to travel abroad with her and leave her free to study. She found one in 1868 in Vladimir Kovalevskij, a paleontologist and publisher of progressive scientific texts. The arrangement was essentially platonic, conducted with the approval of both families, and gave Sofia the legal cover to leave Russia in 1869. She enrolled at the University of Heidelberg, then continued to Berlin to study under the analyst Karl Weierstrass, who initially refused to teach a woman and was talked into private tutorials.

In 1874, she defended a doctoral dissertation at Göttingen summa cum laude — three papers, including the first published version of what is now the Cauchy-Kovalevskaya theorem on partial differential equations. She was the first woman in modern times to earn a PhD in mathematics in the proper sense. The European academic system had no path forward for her; she returned to Russia and spent five years unable to find a teaching position. The Cauchy-Kovalevskaya results would eventually go into every textbook.

Her career resumed in 1883, when the Swedish mathematician Gösta Mittag-Leffler offered her a position at Stockholm's new university. She was promoted to ordinary professor in 1889 — the first European woman in modern times to hold a full mathematics chair. The previous year, in 1888, she had won the French Academy of Sciences' Bordin Prize for an essay on the rotation of a heavy rigid body around a fixed point — a problem the Academy had set as the year's competition. The judges, not knowing the author, doubled the prize money on receiving her solution. She died of pneumonia in 1891 at 41, having become the third completely integrable rigid body, the Kovalevskaya top, in classical mechanics.

In January 1983, Apple hired a 28-year-old artist named Susan Kare to design icons and fonts for a project called Macintosh. Kare had a doctorate in fine art, had been sculpting on commission, and had taught at the San Francisco Art Institute. She had never worked in computer graphics. The business card she had printed for herself read "Macintosh Artist."

The icons she made for the original Mac, which shipped in January 1984, are still in use as design references. The smiling computer that greets you at boot. The trash can. The watch cursor that means waiting. The paint bucket. The lasso. The pointing-hand icon. The bomb that came up when the system crashed. The Command-key clover, lifted from a Swedish road-sign symbol for an interesting site.

Kare worked on a 32-by-32 pixel grid. She bought a notebook of the smallest graph paper she could find at the University Art store in Palo Alto for $2.50 and laid each icon out by filling in squares with a pencil. Her reference points were not earlier computer graphics — there were almost none — but mosaics, needlepoint, and cross-stitch. "Bitmap graphics," she has said, "are like mosaics and needlepoint and other pseudo-digital art forms, all of which I had practised before going to Apple."

The constraint was the medium. A 32-pixel square has 1,024 cells, each of them either on or off. The artist's job was to make the result legible at thumbnail size and unambiguous in meaning. Kare spent days on the trash can, weeks on the bomb.

She left Apple in 1986 to design icons for Microsoft Windows 3.0, IBM OS/2, and later Facebook's gift shop. The trash she drew at Apple in 1983 is still on every Mac.

On the evening of May 22, 1844, a 24-year-old merchant from Shiraz named Sayyid 'Ali Muhammad announced to a young theology student that he was the Báb — "the gate" — through whom the next great prophet of God would soon arrive. The Báb's movement, originally a reformist current within Twelver Shia Islam, gathered tens of thousands of adherents across Persia within six years. The Qajar government and the Shia clerical establishment found the messianic claims and the rapid mobilization unacceptable. The Báb was tried at Tabriz in July 1850 and executed by firing squad in front of about 10,000 spectators. The first volley failed; he was executed at the second.

In 1853, the Persian government exiled most surviving Bábís to Baghdad, then under Ottoman administration. One of them was Mírzá Husayn-'Alí Núrí, a member of a noble Persian family. In April 1863, in a Baghdad garden the Bahá'ís now call Ridván, he announced he was the prophet the Báb had foretold and took the name Bahá'u'lláh — "Glory of God." The Ottomans moved him from Baghdad to Constantinople, then to Adrianople, then in 1868 to the prison-fortress at Acre on the Palestinian coast. He spent the rest of his life there, dying in 1892, and produced an enormous corpus — over 18,000 distinct works of which only about 8 percent have so far been translated into English.

Bahá'u'lláh's son Abdu'l-Bahá led the community after him. Today the faith claims roughly 7 to 8 million adherents in nearly every country and has its administrative seat at the Universal House of Justice in Haifa, Israel. The Bahá'í community is the largest non-Muslim religious minority in Iran, where it remains officially unrecognized and routinely persecuted.

On August 8, 1900, the German mathematician David Hilbert stood up in front of the International Congress of Mathematicians at the Sorbonne in Paris and laid out a program. He gave ten unsolved problems aloud — time was limited — and published the full list of twenty-three in the conference proceedings shortly afterward. The translation that brought the list into English in 1902 was by the American Mary Frances Winston Newson, the first American woman to receive a PhD in mathematics from a European university. Hilbert's twenty-three were a deliberate prospectus for the field. They covered foundations (Problem 1 was the continuum hypothesis), number theory (Problem 8, the Riemann hypothesis), geometry, and physics.

A century of mathematical labor followed. Roughly half the problems are now considered solved by consensus. Some, like Problem 1, were resolved unexpectedly: Paul Cohen showed in 1963 that the continuum hypothesis is independent of the standard axioms of set theory — neither provable nor disprovable from them. Several remain open, the Riemann hypothesis most famously. Hilbert reportedly told colleagues that, if awakened from a thousand-year sleep, his first question would be whether anyone had proved it.

The surprising final note arrived in 2000. The German mathematical historian Rüdiger Thiele, going through Hilbert's working notebooks at Göttingen for a centennial paper, found a draft entry for a twenty-fourth problem that Hilbert had crossed out before publication. The problem asked for a rigorous theory of "simplicity" of mathematical proofs — a way to mathematically rank one proof of a theorem as cleaner than another. Hilbert apparently decided the formulation wasn't sharp enough to be a proper problem and dropped it. Modern proof-complexity theory has come around to it on its own.

A Klein bottle is, in topology, the simplest example of a closed non-orientable surface — a surface that has no inside and no outside, no boundary, and no consistent definition of "left." Walk along it once, all the way around, and you'll find yourself flipped left-for-right at the end. The Klein bottle was first described by the German mathematician Felix Klein in 1882. He coined the German term Kleinsche Fläche — "Klein's surface." The story most topologists tell is that an early translator or typesetter misread Fläche ("surface") as Flasche ("bottle"), and the English-speaking world inherited an apparently unmotivated bottle. Klein's published name was probably surface; the name we use is, almost certainly, a centuries-old typo.

The most striking feature is that the Klein bottle cannot be embedded in three-dimensional space without intersecting itself. The familiar glass-blown sculpture you've seen — with the neck looping through the side wall — is a Klein bottle in 3D immersed, but the self-intersection is real, not a representational shortcut. To get a self-intersection-free Klein bottle you need four spatial dimensions; in 4D you can lift the neck around the wall instead of through it.

The surface plays a particular role in coloring problems. The four-color theorem states that any flat map can be properly colored with four colors. The Heawood conjecture extends this to surfaces of higher genus, and predicts that maps on a Klein bottle would require seven. This turns out to be the only case where Heawood's formula fails: maps on a Klein bottle can always be properly colored with six. Topologically, the Klein bottle is also the connected sum of two real projective planes; cut it along the right plane and it falls apart into two mirror-image Möbius strips.

Joseph Smith Jr. founded the Church of Jesus Christ of Latter-day Saints in 1830 at the age of 24, publishing the Book of Mormon — which he described as a translation, by divine instrument, of writing on golden plates buried near his upstate New York home and revealed to him by an angel in 1823. The text covered the religious history of pre-Columbian American peoples descended from Israelite migrants. The publication of the book gathered, fairly quickly, several thousand converts in New York, Ohio, and Missouri.

The community moved several times under pressure from neighbors. Smith founded the city of Nauvoo on a bend of the Mississippi in Illinois in 1839; under his mayoralty it grew to roughly 12,000 people, briefly the second-largest city in the state. Inside the marriage practices of the inner circle, Smith was secretly entering plural marriages — about thirty to forty additional women across the early 1840s, ten of whom were between 14 and 20. The doctrine was not publicly acknowledged until after his death.

In 1844, Smith launched a campaign for U.S. president. The same year, the dissident Nauvoo Expositor published an exposé of the polygamy. Smith, as mayor, had its printing press destroyed. The destruction of the press triggered legal charges. He surrendered to authorities at Carthage, Illinois. On the afternoon of June 27, 1844, an armed mob of about 200 men with blackened faces stormed Carthage Jail. Smith fired three shots from a smuggled "pepper-box" pistol, wounding three attackers, before being shot multiple times and falling out of an upstairs window. He died at 38. His brother Hyrum was killed in the same attack. Smith remains the first U.S. presidential candidate to have been assassinated.

In 1970, the Container Corporation of America — a Chicago-based paperboard maker — sponsored a graphic-design contest to promote paper recycling and to mark the first Earth Day, held that April 22. Entry was open only to high-school and college students. The brief: design a symbol that would work in black and white, reproduce legibly at a quarter inch, and be free to use forever.

A 23-year-old graduate student in urban planning at the University of Southern California, Gary Anderson, was one of about 500 entrants. He sat down with a pencil, thought about the mechanics of a printing press he had watched and Escher's optical-illusion drawings, and produced three fat triangular arrows folded into a Möbius strip — three arrows chasing each other around a triangle, bent so the loop had a single twist. He posted the entry off and went back to school.

The judges, who included Saul Bass, Herbert Bayer, James Miho, Herbert Pinzke, and Eliot Noyes, picked it. The award was announced at the International Design Conference at Aspen later that year. Anderson got a $2,500 scholarship and a fellowship to attend the conference. Container Corporation released the symbol into the public domain immediately.

It is now one of the most reproduced graphic devices on Earth, printed on packaging in nearly every country with a recycling stream. Anderson, who did not become a graphic designer — he went into urban planning and worked for years on city development around the world — left the symbol off his résumé for decades. He has said in interviews that he assumed it had been forgotten.

In the spring of 325 AD, the emperor Constantine summoned the first ecumenical council of the Christian church to the Bithynian city of Nicaea — modern İznik, Turkey. The empire was a recently Christian one. Constantine had granted toleration with the 313 Edict of Milan, but the church his patronage was now consolidating was visibly fractured along multiple theological lines. The most explosive was a dispute that had begun in Alexandria around 318: the presbyter Arius taught that Christ, while divine, was a created being subordinate to the Father, and his bishop Alexander argued the opposite. Constantine, who cared less about the metaphysics than the cohesion of his own state church, paid for travel, lodging, and meals for as many bishops as cared to attend, and convened them in late spring 325. Tradition gives the count as 318 — a number convenient because it matches the household of Abraham in Genesis. Modern estimates put the actual count between 250 and 300. The council ran from late May until the end of July.

The council's main work was the doctrine. After lengthy debate it produced the Nicene Creed, which described Christ as "begotten not made, of one substance with the Father" — language carefully chosen to exclude Arius. Two of Arius's supporters refused to sign and were exiled with him to Illyria. The council also separated the calculation of Easter from the Jewish Passover, which had been the previous practice in many eastern churches, and issued twenty new disciplinary canons.

What the council did not do is settle the issue. Arianism survived for several centuries; later Roman emperors, including Constantine's own son Constantius II, embraced versions of it. The Nicene Creed itself was substantially revised at a second council in Constantinople in 381, and the version most Christians today recite as "the Nicene Creed" is actually the 381 revision.